What is the integral from 1 to 5 of (ln(x))/(x^5) ?

The integral from 1 to 5 of (ln(x))/(x^5) is 39/625-1/2500 ln(5)

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integral from 1 to 5 of (ln(x))/(x^5)

\int_{1}^{5} \frac{ln ( x )}{x ^{5}} d x

\frac{39}{625} - \frac{1}{2500} ln (5)

Decimal

0.06175 \dots

Solution steps

\int_{1}^{5} \frac{ln ( x )}{x ^{5}} d x

Apply Integration By Parts

= [- \frac{ln ( x )}{4 x ^{4}} - \int - \frac{1}{4 x ^{5}} d x]_{1}^{5}

\int - \frac{1}{4 x ^{5}} d x = \frac{1}{16 x ^{4}}

= [- \frac{ln ( x )}{4 x ^{4}} - \frac{1}{16 x ^{4}}]_{1}^{5}

Compute the boundaries:

\frac{39}{625} - \frac{1}{2500} ln (5)

= \frac{39}{625} - \frac{1}{2500} ln (5)

What is the integral from 1 to 5 of (ln(x))/(x^5) ?
The integral from 1 to 5 of (ln(x))/(x^5) is 39/625-1/2500 ln(5)